[FOM] Intrinsic interest of bounds unfazed

Harvey Friedman friedman at math.ohio-state.edu
Sun Apr 16 21:13:43 EDT 2006

Some of the mathematics Stolzenberg quoted, in his original posting, was
actually from the quote I made of stuff on the Internet, and not from me.
But this is not a significant point in this present posting.


The exchange with Stolzenberg concerning the interest of bounds originally
arose as a response by Stolzenberg to my message Classical/Constructive


Stolzenberg replied to this with


and then the exchange ensued.

With some frequency now, I find myself in exchanges that degenerate into
messages that shed no light on f.o.m. or any intellectual issues. In these
exchanges, I often spend a lot of time researching texts and reporting on
what I find, and presenting careful - and sometimes novel - formulations.

Since I am involved in multiple topics on the FOM, and major research
programs in f.o.m., I have no choice but to sometimes cut off these
exchanges with a summary of my position - often with an attempt to steer the
discussion into productive channels. This very likely will elicit a response
that will go UNANSWERED by me.

HOWEVER, in such cases, if I see significant subscriber interest in the
continuation of the discussion, I will likely continue.

There is nothing in this exchange that casts even the remotest doubt on 1-3

1. The intrinsic interest of the existence of an effective bound for any
particular Pi03 theorem, assuming that the interest of the Pi03 theorem
itself is not at issue.

2. The wide recognition by mathematicians of 1.

3. The intrinsic interest of the existence of primitive recursive bounds, of
indefinitely iterated exponential bounds, of definitely iterated exponential
bounds, of polynomial bounds, of linear bounds, and a whole host of other
categories of bounds to extensive to list, for any particular Pi02 theorem,
assuming that the interest of the Pi02 theorem is not at issue.

4. I have made no effort here to analyze just what makes one bound
sufficiently better than another bound in order to allow publication, or
substantial recognition, or whatever. I will discuss this matter only if I
sense serious interest in it from FOM subscribers.

5. I have spent a fair amount of time citing excerpts from a book of Alan
Baker and a paper of Efim Zelmanov, in addition to the mentioning of famous
work of Faltings and Roth. In the first two cases, the bounds were for Pi02
sentences, starting with Ackerman level bounds, and then they were, over
many years of hard work, knocked down to lower and lower levels. In the
second two cases, the theorems are Pi03, and the existence of recursive
bounds is still wide open and deeply interesting to mathematicians. I also
discussed the Skewes number situation, also a glorious example of deep
interest, where an iterated exponential bound for a single number is
gradually reduced, after a lot of hard work, to something around 10^340,
with more serious efforts ongoing.

FURTHERORE, I find this exchange unproductive in another dimension. Here is
the relevant preamble to my earlier posting


which may have been forgotten:

"The effective bound situation is definitely the UNIQUE place we can point
to now, at which proof theorists, f.o.m. people, constructivists, and core
mathematicians can have serious common grounds and interests. This is why I
continue the exchange concerning attitudes of core mathematicians. The
interest among core mathematicians is clearly sufficient to maintain dialogs
and serious interactions. This is a very good thing, and minimizing it would
serve no useful purpose."

So I hope that subscribers will reread my original posting
Classical/Constructive Arithmetic


and start a productive thread.

Harvey Friedman

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